Academic journal article Education

High and Low Visualization Skills and Pedagogical Decision of Preservice Secondary Mathematics Teachers

Academic journal article Education

High and Low Visualization Skills and Pedagogical Decision of Preservice Secondary Mathematics Teachers

Article excerpt

Introduction

Preservice teacher education programs must provide adequate preparations both in the subject matter and teaching methods. Zazkis and Campbell( 2001) states that" Number theory is a powerful context for conducting research for different areas" (p. 8). According to Zazkis and Campbell (2002) "Elementary concepts of number theory, despite their importance to the field of mathematics, have received scant attention in mathematics education research." (p.540). Selden and Selden goes further "Number Theory appears to be a rather neglected area in the mathematics education research literature" (p.213). In mathematics visualization is central to many mathematical concepts (Cunningham 1994; Presmeg, 2006; Zarzycki, 2004;), Zarzycki (2004) points out that "We could not even imagine introducing many mathematical concepts without illustrating them by pictures, drawings, graphs, etc." (p.108).

Many teachers are not prepared to implement instructional strategies that are grounded in high curriculum standards (Elmore & Burney, 1996; Grant, Peterson, & Shojgreen-Downer, 1996). The shift in emphasis on mathematical understanding (as compared to memorization facts) means that teachers must learn more about mathematics as well as how students learn this mathematics. Garet, Porter, Desimone, Birman, & Yoon (2001) describe structural and core features of effective professional development. Structural features focus on the types of activities (e.g., workshops, institutions, courses); duration; and collective participation. Core features are focusing on content, promoting active learning, and fostering coherence. The inclusion of a focus on content in professional development activities is seen to vary across four dimensions. These are the degree of emphasis on subject matter and teaching methods, the specificity of change in teaching practice encouraged, the degree of emphasis on goals for student learning, and the degree of emphasis on the ways students learn. Garet, et.al, (2001) describes the second core feature of promoting active learning as one that "concerns the opportunities provided by the professional development activity for teachers to become actively engaged in meaningful discussion, planning, and practice" (p. 925). The third core feature of fostering coherence is concerned with "the extent to which professional development activities are perceived by teachers to be a part of a coherent program of teacher learning" (p. 927).

This report attempts to explore the beauty and richness of viewing number theory problems from a visual-arithmetic perspective. Studies (e.g., Vinner, 1989; Tall, 1991; Eisenberg, 1994) have consistently shown that students' understanding is typically analytic and not visual. Three possible reasons for this are when the analytic mode, instead of the graphic mode, is pervasively used in instruction, or when students or teachers hold the belief that mathematics is the skillful manipulation of symbols and numbers and finally visual proof as not valued or accepted as proof in mathematics. It is clear from the literature (e.g., Lesh, Post, & Behr, 1987; Janvier, 1987; NCTM, 2000) that having multiple ways - for example, graphic and analytic - to represent mathematical concepts is beneficial

Our contention is not that one student's representational scheme is superior to another, only that students often construct vastly different personal and idiosyncratic representations which lead to different understandings of a concept. Because student-generated representations provide useful windows into students' thinking, it is productive for teachers to value these personal representations. Moreover, there is a belief among mathematics educators (e.g., Janvier 1987; Lesh, Post, & Behr, 1987) that students benefit from being able to understand a variety of representations for mathematical concepts and to select and apply a representation that is suited to a particular mathematical task. …

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